While I'm on the topic of induction, I should discuss its relation to the philosophy of "falsification". Or rather, its opposition to said philosophy.
Falsification is perhaps the most well-known piece of philosophy of science. The idea was invented by Karl Popper around the 1930s. Among other things, it was meant to answer the demarcation problem, the question of what is and isn't science. A theory is scientific if it is falsifiable; it is unscientific if it is unfalsifiable. By "falsifiable," we mean that there is some piece of evidence that might disprove the theory. If we have the theory, "All crows are black," this can be falsified by the observation of a white crow. The reasoning behind this piece of philosophy is that you can never prove that all crows are black, at least not in practical terms. But we can disprove it, by observing a white crow. So instead of trying to prove it, we should simply try our best to disprove it.
Despite being the popular view of how to distinguish science from non-science, falsifiability is not really how most scientists themselves view it. This is because science doesn't actually work that way, not exactly. Nor is it apparent that it should work that way. Scientists don't exclusively spend their time trying to disprove their own ideas. But my criticism comes from a different direction.
My problem with falsification is that it buys into the dichotomy between "positive" and "negative" claims. It's said you can't prove negative claims (ie, the non-existence of a particular object) but you can prove positive claims (ie, the existence of a particular object). While this certainly describes a lot of different claims, in general, there is no dichotomy. It's not necessarily easier to prove positive claims than negative ones. After all, the distinction between positive and negative is artificial. Any positive claim "P" can be made into the negative claim "not-(not-P)".
For example, consider the claim, "More peppered moths are black than white." You can't disprove this by simply finding a white peppered moth. Nor can you prove it by finding a black peppered moth. In fact, you can't ever absolutely disprove or prove it! You can come pretty close by observing a large random sampling, but you never prove or disprove anything.
More sophisticated forms of falsification account for this by saying you can falsify a theory when the evidence is so great that it's no longer reasonable. But no one single observation can falsify the theory, so when exactly does it go from unfalsified to falsified? Wouldn't it be more useful to be able to characterize all the grays between proof and disproof (especially when neither extreme is actually possible), or perhaps even quantify them?
An alternative to falsification is inductionism. Induction does not purport to be able to prove or disprove anything. But it can argue that certain claims are more or less likely, and that can be almost as good as proof. There is even some mathematical underpinning to it, so you could, in principle, quantify your grays. There are a few assumptions made, but they are not unreasonable, and we can always make exceptions for those few circumstances in which the assumptions are questionable.
And of course, the third alternative is to accept both inductionism and falsification. I think Popper saw falsification as a replacement for induction, not a supplement, but who am I to let Popper dictate our options? The problem is that falsification is usually more or less the same as induction, only less powerful. Other times, it seems exactly the same, except with clunkier terminology. The only time I think falsification is useful is in its simple solution to the demarcation problem. It makes distinguishing science from non-science easy. But then, I think it can be wrong sometimes, because it is too simplistic. Perhaps there are some scientific claims that can't be falsified, or unscientific claims that can be falsified.
Perhaps I can't put the final nail in the coffin of falsification, but I intend this for a general audience that perhaps has not previously questioned Popper's ideas about science. The take-home message is that falsification is not a universally accepted way to think about science, and should not be taken for granted. Usually, there is never any particular point where a scientific theory is clearly falsified, but that doesn't mean we can't make progress.